Finite Group Databasebeta

← #11713235

Subgroups — #11713235

Order 768.

554 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 57 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊(#41⋊C3)C211yesmaximal, characteristic
2C2×#5426C211yesmaximal, characteristic, complemented
3#73151C212yesmaximal, complemented
4#73151C211yesmaximal, characteristic, complemented
5#73059C212yesmaximal, complemented
6#24045—31nomaximal, Sylow
7C22×#41⋊C3C2211yescharacteristic
8C22×#41⋊C3C2211yescharacteristic
9C4⋊(#41⋊C3)—22no—
10C2×C4×SL(2,𝔽3)C2211yescharacteristic
11#5426—22no—
12C2×#859C2212yes—
13C2×#859C2212yes—
14#4981—41nomaximal
15C2×#550D311yescharacteristic
16C2×#481—31no—
17C2×#417—31no—
18#2486—31no—
19#2041—32no—
20#2034—32no—
21#1895—31no—
22#1870—31no—
23#1865—32no—
24#1788—32no—
25#1788—31no—
26C2×#41⋊C3—21no—
27C2×#41⋊C3—22no—
28C2×#41⋊C3—22no—
29C2×#41⋊C3D412yes—
30C22×SL(2,𝔽3)C2311yescharacteristic
31C2×C3×C4⋊C4—41no—
32C2×C4⋊Dic3—41no—
33C2×Dic3⋊C4—42no—
34C4×SL(2,𝔽3)—21no—
35#859—22no—
36#859—22no—
37C3⋊#95—42no—
38C3⋊#95—41no—
39C22×#41C2×D311yescharacteristic
40C22×#41C2×D311yescharacteristic
41C22×M16—31no—
42#550—22no—
43#550—62no—
44C2×#127—31no—
45C2×#123—61no—
46C2×#121—31no—
47C2×#121—31no—
48C2×C4×Q8C2×D311yescharacteristic
49C2×C4×D4—31no—
50C22×C4⋊C4—31no—
51#481—64no—
52C2×#106—32no—
53#417—62no—
54#417—62no—
55C2×#103—32no—
56C2×#103—32no—
57C2×#102—32no—
58C2×#102—32no—
59C2×#102—32no—
60C2×#102—32no—
61C2×C22⋊C8—31no—
62C2×C22⋊C8—32no—
63#398—61no—
64#397—61no—
65#389—32no—
66#381—61no—
67#380—31no—
68#380—32no—
69#340—32no—
70#336—32no—
71C22×C3×C4—41no—
72C22×C3×C4—41no—
73C22×C3×C4—41no—
74C22×Dic3—42no—
75C22×Dic3—42no—
76#41⋊C3—22no—
77#41⋊C3—41no—
78C2×SL(2,𝔽3)C2×D411yescharacteristic
79C2×SL(2,𝔽3)#4112yes—
80C3×C4⋊C4—82no—
81C4⋊Dic3—82no—
82Dic3⋊C4—84no—
83C2×#41—22no—
84C2×#41—22no—
85C2×#41—21no—
86C2×#41—32no—
87C2×#41—61no—
88C2×#41—62no—
89C2×#41—62no—
90C2×#41#6712yes—
91C22×Q8C22×D311yescharacteristic
92C22×D4—31no—
93C22×D4—31no—
94C22×D4—31no—
95C22×D4—61no—
96C23×C4—31no—
97C23×C4—31no—
98C23×C4—61no—
99C2×M16—62no—
100C2×M16—122no—
101C22×C8—32no—
102C22×C8—32no—
103#127—62no—
104#127—62no—
105#123—124no—
106#121—34no—
107#121—62no—
108#121—62no—
109#121—62no—
110C4×Q8—21no—
111C4×Q8—61no—
112C4×D4—62no—
113C4×D4—62no—
114C2×C4⋊C4—31no—
115C2×C4⋊C4—32no—
116C2×C4⋊C4—32no—
117C2×C4⋊C4—62no—
118C2×C4⋊C4—62no—
119C2×C4⋊C4—61no—
120C2×C4⋊C4—62no—
121C2×C4⋊C4S411yescharacteristic
122C2×C22⋊C4—31no—
123C2×C22⋊C4—31no—
124C2×C22⋊C4—61no—
125C2×C22⋊C4—62no—
126C2×C42—31no—
127#106—64no—
128#103—62no—
129#103—62no—
130#103—62no—
131#103—62no—
132#102—64no—
133#102—64no—
134#102—122no—
135#102—122no—
136C22⋊C8—62no—
137C22⋊C8—122no—
138#95—62no—
139#95—62no—
140#95—61no—
141C23×C3—41no—
142C2×C3×C4—42no—
143C2×C3×C4—42no—
144C2×C3×C4—81no—
145C2×C3×C4—82no—
146C2×C3×C4—81no—
147C2×C3×C4—81no—
148C2×C3×C4—82no—
149C2×Dic3—44no—
150C2×Dic3—82no—
151C2×Dic3—82no—
152C2×Dic3—82no—
153C2×Dic3—82no—
154C2×Dic3—82no—
155SL(2,𝔽3)#12311yescharacteristic, complemented
156C24—31no—
157C24—31no—
158#41—22no—
159#41—41no—
160#41—62no—
161#41—64no—
162#41—122no—
163#41—122no—
164#41—122no—
165#41—121no—
166C2×Q8—31no—
167C2×Q8—32no—
168C2×Q8#23712yes—
169C2×Q8C2×#6711yescharacteristic
170C2×D4—32no—
171C2×D4—32no—
172C2×D4—62no—
173C2×D4—62no—
174C2×D4—62no—
175C2×D4—62no—
176C2×D4—62no—
177C2×D4—61no—
178C2×D4—61no—
179C2×D4—64no—
180C2×D4—62no—
181C2×D4—121no—
182C2×D4—242no—
183C22×C4—31no—
184C22×C4—32no—
185C22×C4—32no—
186C22×C4—31no—
187C22×C4—62no—
188C22×C4—62no—
189C22×C4—62no—
190C22×C4—61no—
191C22×C4—61no—
192C22×C4—62no—
193C22×C4—62no—
194C22×C4—62no—
195C22×C4—122no—
196C22×C4—122no—
197C22×C4—122no—
198C22×C4C2×S411yescharacteristic
199C22×C4C2×S411yescharacteristic
200C22×C4C2×S411yescharacteristic
201M16—124no—
202C2×C8—62no—
203C2×C8—62no—
204C2×C8—62no—
205C2×C8—62no—
206C2×C8—62no—
207C2×C8—62no—
208C4⋊C4—22no—
209C4⋊C4—32no—
210C4⋊C4—61no—
211C4⋊C4—62no—
212C4⋊C4—62no—
213C4⋊C4—62no—
214C4⋊C4—62no—
215C4⋊C4—124no—
216C4⋊C4—122no—
217C22⋊C4—62no—
218C22⋊C4—62no—
219C22⋊C4—122no—
220C22⋊C4—124no—
221C42—61no—
222C42—61no—
223C22×C3—42no—
224C22×C3—41no—
225C22×C3—41no—
226C22×C3—42no—
227C22×C3—41no—
228C3×C4—82no—
229C3×C4—82no—
230C3×C4—161no—
231Dic3—84no—
232Dic3—162no—
233C23—32no—
234C23—62no—
235C23—61no—
236C23—61no—
237C23—62no—
238C23—62no—
239C23—61no—
240C23—61no—
241C23—62no—
242C23—61no—
243C23C22×S411yescharacteristic
244Q8—32no—
245Q8—31no—
246Q8—61no—
247Q8#93511yescharacteristic
248D4—62no—
249D4—62no—
250D4—64no—
251D4—124no—
252D4—122no—
253D4—121no—
254D4—121no—
255D4—244no—
256C2×C4—22no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—21no—
261C2×C4—32no—
262C2×C4—32no—
263C2×C4—31no—
264C2×C4—31no—
265C2×C4—62no—
266C2×C4—62no—
267C2×C4—64no—
268C2×C4—62no—
269C2×C4—61no—
270C2×C4—121no—
271C2×C4—121no—
272C2×C4—122no—
273C2×C4—122no—
274C2×C4—122no—
275C2×C4—122no—
276C2×C4—122no—
277C2×C4—122no—
278C2×C4—122no—
279C2×C4—244no—
280C2×C4#98612yes—
281C2×C4GL(2,ℤ/4)12yes—
282C8—122no—
283C8—122no—
284C2×C3—41no—
285C2×C3—41no—
286C2×C3—42no—
287C2×C3—42no—
288C2×C3—41no—
289C22—62no—
290C22—62no—
291C22—62no—
292C22—122no—
293C22—122no—
294C22—121no—
295C22—121no—
296C22—122no—
297C22—122no—
298C22—121no—
299C22#591412yes—
300C22C2×#98611yescharacteristic
301C22#592311yescharacteristic
302C22#592412yes—
303C22C2×GL(2,ℤ/4)11yescharacteristic
304C4—22no—
305C4—22no—
306C4—31no—
307C4—32no—
308C4—31no—
309C4—41no—
310C4—121no—
311C4—242no—
312C4—242no—
313C3—41noSylow
314C2—122no—
315C2—241no—
316C2#8540512yes—
317C2#8541212yes—
318C2#8549111yescharacteristic
319C2#8552311yescharacteristic
320C2#8552711yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, cosocleC22×SL(2,𝔽3)C23
FittingC2×#550D3
prime radicalC2×C4⋊(#41⋊C3)C2
hypercenterC2×C4⋊C4S4
hypocenterSL(2,𝔽3)#123
LCSG ▷ C22×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#5426C2semidirect
3. #73151C2semidirect
4. #73151C2semidirect
5. #73059C2semidirect
155. SL(2,𝔽3)#123semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×C4⋊(#41⋊C3)2
C2×#54262
#731512
#730592
#240453
#49814